Unified Field Theory, Riemann Hypothesis, Euler-Mascheroni constant

1. The Navier-Stokes equations (NSE) described in a nutshell by AI

In the Navier-Stokes equations, the term involving static pressure (P), denoted as -∇P, represents the forces (*) acting on a fluid parcel due to normal stresses. Specifically, it signifies the effect of internal pressure acting on a fluid particle, causing it to be pushed by pressure from surrounding fluid parcels. This pressure term is a fundamental component of the forces that influence fluid motion, resisting compression and playing a crucial role in momentum balance for Newtonian fluids.

(*) Note: this is where AI is wrong; ∇P represents the acceleration of a mechanical fluid into the direction of its lower pressure state (whatever the fluid gets this information from). In terms of the central concepts of building block 1 the "pressure force" phenomenon is explained by the potential difference between the mechanical fluid particle and its intrinsic dynamic anti-particle, defined by a self-adjoint dynamic potential operator. In terms of the central concepts of the building block 2 the "pressure force" phenomenon is explained by the difference of the following two potentials: the mechanical potential of the fluid defined by the self-adjoint Friedrichs extension of the Laplacian operator with H(1) domain, and the dynamic potential of the fluid defined by a "complementary" compact Schrödinger 2.0 operator with H(1,ortho) domain.

2. Problem statements

The Navier-Stokes Equations (NSE) describes a flow of incompressible, viscous fluid. The three key foundational questions of every PDE is existence, and uniqueness of solutions, as well as whether solutions corresponding to smooth initial data can develop singularities in finite time, and what these might mean. For the NSE satisfactory answers to those questions are available in two dimensions, i.e. 2D-NSE with smooth initial data possesses unique solutions which stay smooth forever. In three dimensions, those questions are still open. Only local existence and uniqueness results are known. Global existence of strong solutions has been proven only, when initial and external forces data are sufficiently smooth. Uniqueness and regularity of non-local Leray-Hopf solutions are still open problems. The question of global existence of smooth solutions vs. finite time blow up is one of the millennium problems of the Clay Mathematics Institute.

It has been questioned whether the NSE really describes general flows: The difficulty with ideal fluids, and the source of the d'Alembert paradox, is that in such fluids there are no frictional forces. Two neighboring portions of an ideal fluid can move at different velocities without rubbing on each other, provided they are separated by a streamline. It is clear that such a phenomenon can never occur in a real fluid, and the question is how frictional forces can be introduced into a model of a fluid.

Navier-Stokes Equation - Clay Mathematics Institute

"This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations."

3. The paradigm change

The average velocity of the fluids in a fluid continuum are elements of the mechanical Hilbert space H(1) accompanied by fluid particles as elements of the (thermo-statistical) Hilbert space L(2). The additional dynamic energy field of building block 2 is represented by the closed sub-space of H(1/2) complementary to the mechanical energy space H(1).

The current variational NSE system is based on the Newtonian flow, which is governed by two types of forces: the volume forces (the gravity force and viscous forces) and the surface pressure force. The two unknown function of the NSE system are the fluid velocity u and the pressure p. The overall principle to define the NSE system is the Newton principle F=m*a. In case of the pressure force the related acceleration a is given by the gradient of the pressure.

In the H(1/2) based total energy Hilbert space framework the volume forces are still governed by the mechanical energy, while the surface pressure force is newly governed by the closed sub-space of H(1/2) complementary to the H(1). This sub-space also provides an appropriate modelling framework of surface frictional forces.

The technical tools to handle surface forces in a H(1/2) energy Hilbert space framework of the related Neumann equation problems are

1. Plemelj's extended single layer potential operator
the mass density function is replaced by a "mass element" on the surface, which makes the concept of a normal obsolete

2. the Prandtl double layer operator

3. the Bogovskii operator in Sobolev spaces of negative order, (GeM)

4. the mechanical energy space is governed by Fourier waves; its complementary closed sub-space in H(1/2) is governed by Calderon wavelets.

Note: The current variational NSE representation in the (thermo-) statistical Hilbert space L(2)=H(0) accompanied by the mechanical energy Hilbert space H(1) represents the Newtonian mechanical world. IT may be interpreted as a Ritz-Galerkin approximation model of the new H(1/2) energy Hilbert space based dynamic fluid NSE system, e.g. (VeW).

4. A global bounded energy norm estimate of the 3D-NSE system enabled by complementary mechanical and turbulence energy spaces

Braun K., A global bounded energy norm estimate of the 3D-NSE system enabled by complementary mechanical and turbulence energy spaces.pdf

5. The current conceptual problem: the purely Newtonian mechanical fluid based solution fields (u,p) ex H(1) x H(0) = L(2)

Applying formally the div-operator to the classical NSE the pressure field must satisfy the interior Neumann problem acompanied by a boundary condition requiring the outward unit normal at the boundary. The initial boundary value problem determines the initial pressure by the Neumann problem for t --> 0. From this it follows that in this framework the prescription of the pressure at the boundary walls or at the initial time independently of the velocity field u, could be incompatible with and, therefore, could retender the problem ill-posed, (GaG).

In (HeJ) a (u,p) solution of the NSE system is provided with a divergence free initial boundary value velocity v(0) ex H(0) = L(2) and a corresponding scalar pressure field p, which is infinite at t=0.

(GaG) Galdi G. P., The Navier-Stokes Equations: A Mathematical analysis, Birkhäuser Verlag, Monographs in Mathematics, ISBN 978-3-0348-0484-4

(HeJ) Heywood J. G., Walsh O. D., A counter-example concerning the pressure in the Navier-Stokes equations, as t à 0, Pacific J. Math., 164 (1994), 351-359.

6. A dynamic fluid artifact with intrinsic „pressure force“

In the current NSE system the pressure plays the key role to generate the „force“ (resp. to provide the energy) to move a fluid particle forwards into the direction of the decreasing pressure. The „pressure“ is a scalar quantity, however, its spacial shift generates a „pressure force“, which acts on the fluid. Correspondingly, the negative pressure gradient represents the acceleration of the fluid which acts on the fluid. Its multiplication with the mass density of the fluid continuum it gives the fundamental force, which governs the movement of fluids orchestrated by the Newton law F = m * a, (e.g. (LaL).

It is the difference between two fluid pressures that generates this force, which moves the considered fluid forward. The negative gradient of this pressure (the relevant term in the NSE system) represents the acceleration of the considered fluid into the direction of the decreasing pressure.

The current pressure force is the product of the mass density of the fluid continuum and the accelearation of the mechanical fluid moving into the direction of the fluid.

The intrinsic „pressure force“ is modelled by the fluid intrinsic dynamic energy. This concept replaces the sophisticated term, given by the fluid intrinsic acceleration (i.e. the gradient of the pressure) divided by the density constant, by a new dynamic fluid flow on the considered surface.

Note: In classical electricity theory the electric flux is a flux of charges transported by the electrons in metals. The „pressure“ by which the electrons are pushed into the conductive wire is called electric tension or electric potential.

Note: The inner product of H(1/2) is isometric to an inner product in the form (Qx,Px), where Q resp. P denote Schrödinger‘s position & momentum operators. The combination with the Riesz transform operator provides the link to the alternatively proposed Schrödinger operator 2.0 in (BrK2).

Note: The extended H(1/2) energy field based variational framework allows ‘optimal’ FEM approximation error estimates for non-linear parabolic problems with non-regular initial value data using the Stefan (free boundary) model problem, (see below).

(DiE) DiBenedetto E., Partial Differential Equations, Birkhäuser, Boston, Basel, New York, 2010

(GeM) Geißert, M., Heck, H., Hieber, M. (2006). On the Equation div u = g and Bogovskii’s Operator in Sobolev Spaces of Negative Order. In: Koelink, E., van Neerven, J., de Pagter, B., Sweers, G., Luger, A., Woracek, H. (eds) Partial Differential Equations and Functional Analysis. Operator Theory: Advances and Applications, vol 168. Birkhäuser Basel

(GeM) Geißert, M., Heck, H., Hieber, M. (2006). On the Equation div u = g and Bogovskii’s Operator in Sobolev Spaces of Negative Order. In: Koelink, E., van Neerven, J., de Pagter, B., Sweers, G., Luger, A., Woracek, H. (eds) Partial Differential Equations and Functional Analysis. Operator Theory: Advances and Applications, vol 168. Birkhäuser Basel

(LiI) Lifanov I. K., Poltavskii L. N., Vainikko G. M., Hypersingular integral equations and their applictions, ChapmAN 6 Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004

(PlJ) Plemelj J., Potentialtheoretische Untersuchungen, B.G. Teubner, Leipzig, 1911

7. The extended H(1/2) based dynamic fluid model alternatively to Kolmogorov’s statistical turbulence model

Quote from W. Heisenberg: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first."

What is turbulence?
from M. Farge, K. Schneider, Wavelets: application to turbulence

„Turbulence is a highly nonlinear regime encountered in fluid flows. They are described by continuous fields, e.g., velocity or pressure, assuming that the characteristic scale of the fluid motions is much larger than the mean free path of the molecular motions. The prediction of the space-time evolution of fluid flows from first principles is given by the solutions of the Navier-Stokes equations. The turbulent regime develops when the nonlinear term of Navier-Stokes equations strongly dominates the linear term; the ratio of the norms of both terms is the Reynolds number Re, which characterizes the level of turbulence. In this regime nonlinear instabilities dominate, which leads to the flow sensitivity to initial conditions and unpredictability. The corresponding turbulent fields are highly fluctuating and their detailed motions cannot be predicted, although, if one assumes some statistical stability of the turbulence regime, averaged quantities, such as mean and variance, or other related quantities, e.g., diffusion coefficients, lift or drag, may still be predicted.

When turbulent flows are statistically stationary (in time) or homogeneous (in space), as it is classically supposed, one studies their energy spectrum, given by the modulus of the Fourier transform of the velocity auto-correlation. Unfortunately, since the Fourier representation spreads the information in physical space among the phase of all Fourier coefficients, the energy spectrum looses all structural information in time or space. This is a major limitation of the classical way of analyzing turbulent flows. This is why we have proposed to use the wavelet representation instead and define new analysis tools able to preserve time or space locality.“

In the proposed extended variational H(1/2) energy Hilbert space the included mechanical Hilbert energy space H(1) (which is based on the statistical L(2)=H(0) Hilbert space) is governed by Fourier waves, while its complementary closed sub-space of H(1/2) may be governed by wavelets.